Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
The entry for infinitesimal extension said that an infinitesimal extension of rings was an epimorphism of rings with nilpotent kernel. I’ve changed this to say a quotient map of rings with nilpotent kernel. I hope this is correct: for example, localization maps are ring epimorphisms, and often have zero kernel (so in particular, nilpotent kernel) but geometrically these correspond to dense open inclusions, which are in no sense infinitesimal extensions.
Tim, thanks for catching this, my bad. Todd is right about what made me make this mistake. I have added a corresponding remark to the entry, following Todd’s pointer.
The finiteness condition is interesting. Perhaps the correct definition is actually “an infinitesimal extension is a finite epimorphism (=finite quotient) of rings with nilpotent kernel”. After all, the definitions of “etale”, “unramified”, and “smooth” all require a finite-type condition that differentiates them from “formally etale”, “formally unramified”, and “formally smooth”. Should one distinguish between “formally infinitesimal” versus “infinitesimal” here? Well – one would have to choose a different term, to avoid clashing with “formal scheme” which is also in the neighborhood.
I suppose the answer should depend on what the theorems are supposed to be, and there’s some latitude – I take it that “infinitesimal extension” isn’t exactly a standard term in the algebraic geometry literature, right?
I take it that “infinitesimal extension” isn’t exactly a standard term in the algebraic geometry literature, right?
That’s true. But for instance Lurie in his “deformation contexts” speaks of “small etensions” for (finite) infinitesimal extensions. It seems to me that “infinitesimal extension” is at least as good a term as “small” in this context.
The question by Eric Finster here at MO may be useful.
I thought that an extension of commutative algebras was infinitesimal if it has square zero kernel. I think I met the idea in Lichtenbaum and Schlessing’s cotangent complex paper and also in Quillen’s one on cohomology of commutative rings. It is certainly used that way in ’Noncommutative Geometry and Cayley-smooth Orders’ by Lieven Le Bruyn which I found on Google.
1 to 6 of 6